Policies.Posterior.Gamma module¶
Manipulate a Gamma posterior. No need for tricks to handle non-binary rewards.
- See https://en.wikipedia.org/wiki/Gamma_distribution#Conjugate_prior
- And https://en.wikipedia.org/wiki/Conjugate_prior#Continuous_distributions
-
class
Policies.Posterior.Gamma.
Gamma
(k=1, lmbda=1)[source]¶ Bases:
Policies.Posterior.Posterior.Posterior
Manipulate a Gamma posterior.
-
__init__
(k=1, lmbda=1)[source]¶ Create a Gamma posterior, \(\Gamma(k, \lambda)\), with \(k=1\) and \(\lambda=1\) by default.
-
k
= None¶ Parameter \(k\)
-
lmbda
= None¶ Parameter \(\lambda\)
-
reset
(k=None, lmbda=None)[source]¶ Reset k and lmbda, both to 1 as when creating a new default Gamma.
-
sample
()[source]¶ Get a random sample from the Beta posterior (using
numpy.random.gammavariate()
).- Used only by
Thompson
Sampling andAdBandits
so far.
- Used only by
-
quantile
(p)[source]¶ Return the p quantile of the Gamma posterior (using
scipy.stats.gdtrix()
).- Used only by
BayesUCB
andAdBandits
so far.
- Used only by
-
update
(obs)[source]¶ Add an observation: increase k by k0, and lmbda by obs (do not have to be normalized).
-
__module__
= 'Policies.Posterior.Gamma'¶
-
-
Policies.Posterior.Gamma.
gammavariate
()¶ gamma(shape, scale=1.0, size=None)
Draw samples from a Gamma distribution.
Samples are drawn from a Gamma distribution with specified parameters, shape (sometimes designated “k”) and scale (sometimes designated “theta”), where both parameters are > 0.
Note
New code should use the
gamma
method of adefault_rng()
instance instead; please see the Quick Start.- shape : float or array_like of floats
- The shape of the gamma distribution. Must be non-negative.
- scale : float or array_like of floats, optional
- The scale of the gamma distribution. Must be non-negative. Default is equal to 1.
- size : int or tuple of ints, optional
- Output shape. If the given shape is, e.g.,
(m, n, k)
, thenm * n * k
samples are drawn. If size isNone
(default), a single value is returned ifshape
andscale
are both scalars. Otherwise,np.broadcast(shape, scale).size
samples are drawn.
- out : ndarray or scalar
- Drawn samples from the parameterized gamma distribution.
- scipy.stats.gamma : probability density function, distribution or
- cumulative density function, etc.
Generator.gamma: which should be used for new code.
The probability density for the Gamma distribution is
\[p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)},\]where \(k\) is the shape and \(\theta\) the scale, and \(\Gamma\) is the Gamma function.
The Gamma distribution is often used to model the times to failure of electronic components, and arises naturally in processes for which the waiting times between Poisson distributed events are relevant.
[1] Weisstein, Eric W. “Gamma Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/GammaDistribution.html [2] Wikipedia, “Gamma distribution”, https://en.wikipedia.org/wiki/Gamma_distribution Draw samples from the distribution:
>>> shape, scale = 2., 2. # mean=4, std=2*sqrt(2) >>> s = np.random.gamma(shape, scale, 1000)
Display the histogram of the samples, along with the probability density function:
>>> import matplotlib.pyplot as plt >>> import scipy.special as sps # doctest: +SKIP >>> count, bins, ignored = plt.hist(s, 50, density=True) >>> y = bins**(shape-1)*(np.exp(-bins/scale) / # doctest: +SKIP ... (sps.gamma(shape)*scale**shape)) >>> plt.plot(bins, y, linewidth=2, color='r') # doctest: +SKIP >>> plt.show()